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Flat knot 6.631

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,2,2,4,2,1,2,2,1,1,-1,1,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.631']
Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369']
Outer characteristic polynomial of the knot is: t^7+58t^5+71t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.631']
2-strand cable arrow polynomial of the knot is: -864*K1**4 + 864*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1184*K1**3*K3 - 2560*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 6624*K1**2*K2 - 1760*K1**2*K3**2 - 128*K1**2*K3*K5 - 64*K1**2*K4**2 - 6592*K1**2 + 96*K1*K2**3*K3 - 672*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7504*K1*K2*K3 + 1888*K1*K3*K4 + 224*K1*K4*K5 - 208*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 632*K2**2*K4 - 4486*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 2760*K3**2 - 608*K4**2 - 72*K5**2 - 2*K6**2 + 4670
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.631']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11003', 'vk6.11082', 'vk6.12173', 'vk6.12280', 'vk6.18205', 'vk6.18542', 'vk6.24667', 'vk6.25091', 'vk6.30572', 'vk6.30667', 'vk6.31846', 'vk6.31893', 'vk6.36799', 'vk6.37255', 'vk6.44042', 'vk6.44384', 'vk6.51814', 'vk6.51881', 'vk6.52682', 'vk6.52776', 'vk6.55999', 'vk6.56274', 'vk6.60538', 'vk6.60880', 'vk6.63498', 'vk6.63542', 'vk6.63980', 'vk6.64024', 'vk6.65664', 'vk6.65948', 'vk6.68712', 'vk6.68922']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4U1O5U2U4O6U5U6U3
R3 orbit {'O1O2O3O4U1O5U2U4O6U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U6O5U1U3O6U4
Gauss code of K* O1O2O3U4U5U3U6O4U1O5O6U2
Gauss code of -K* O1O2O3U2O4O5U3O6U4U1U5U6
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -2 2 1 1 1],[ 3 0 1 3 2 2 0],[ 2 -1 0 3 1 2 1],[-2 -3 -3 0 -1 0 1],[-1 -2 -1 1 0 1 1],[-1 -2 -2 0 -1 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 1 0 -1 -3 -3],[-1 -1 0 -1 -1 -1 0],[-1 0 1 0 -1 -2 -2],[-1 1 1 1 0 -1 -2],[ 2 3 1 2 1 0 -1],[ 3 3 0 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,-1,0,1,3,3,1,1,1,0,1,2,2,1,2,1]
Phi over symmetry [-3,-2,1,1,1,2,0,2,2,4,2,1,2,2,1,1,-1,1,-1,0,2]
Phi of -K [-3,-2,1,1,1,2,0,2,2,4,2,1,2,2,1,1,-1,1,-1,0,2]
Phi of K* [-2,-1,-1,-1,2,3,0,1,2,1,2,1,1,2,2,1,1,2,2,4,0]
Phi of -K* [-3,-2,1,1,1,2,1,0,2,2,3,1,1,2,3,-1,-1,-1,1,1,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 5z^2+25z+31
Enhanced Jones-Krushkal polynomial 5w^3z^2+25w^2z+31w
Inner characteristic polynomial t^6+38t^4+17t^2
Outer characteristic polynomial t^7+58t^5+71t^3+6t
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -864*K1**4 + 864*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1184*K1**3*K3 - 2560*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 6624*K1**2*K2 - 1760*K1**2*K3**2 - 128*K1**2*K3*K5 - 64*K1**2*K4**2 - 6592*K1**2 + 96*K1*K2**3*K3 - 672*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7504*K1*K2*K3 + 1888*K1*K3*K4 + 224*K1*K4*K5 - 208*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 632*K2**2*K4 - 4486*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 2760*K3**2 - 608*K4**2 - 72*K5**2 - 2*K6**2 + 4670
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
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